$\newcommand{\Xc}{\mathcal{X}}$ Let $\Xc=[-1,1]$ and consider a Gaussian kernel $k(x,t)\propto \exp(-(x-t)^2/2\sigma^2)$ for some $\sigma>0$ on $\Xc$. I am looking for an eigenfunction expansion of the induced integral operator $\mathbf{K}\colon L^2(\Xc)\to L^2(\Xc)$. That is, I wish to find a countable set of functions $\{\phi_n\colon \Xc\to \mathbb{R}\}_{n=1}^\infty$ such that $$ k(x,t) = \sum_{n=1}^{\infty} \lambda_n \phi_n(x)\phi_n(t) $$ and $\int_{\Xc}\phi_m(x)\phi_n(x)d x=\delta_{mn}$.
Here are some related results I am aware of:
- Hermite polynomials characterize the $L^2(\rho)$-orthonormal eigenfunctions of Gaussian kernels over $\mathbb{R}$ for a Gaussian weight $\rho$ [See Sec. 6.2].
- Spherical harmonics characterize the $L^2(\Xc)$-orthonormal eigenfunctions of Gaussian kernels over the hypersphere $\Xc=S^{d-1}$ (for a uniform weight) [See Thm. 2.].